Special equipment: A cookie sheet or jelly roll pan covered with parchment or foil; two 9-inch round cake pans
For the tart crust, roll the dough to a rough 12-inch disk and transfer it to the cookie sheet. Pierce the dough all over with a fork and place a plate or pattern on to and cut it to a perfect 11-inch circle. Butter the bottom of one of the cake pans and center it on the disk of dough. Chill one hour.
While the dough is chilling, cook the apricots. Preheat the oven to 375 degrees and set a rack in the middle level. Thickly butter the other pan and sprinkle it with the brown sugar. Arrange the apricot halves, skin side down in the pan, alternating a dried apricot with every two fresh apricot halves. Sprinkle with the remaining butter and then the lemon juice. Bake the apricots about 40 minutes, until they are well cooked through. Cool on a rack.
To bake the tart crust, lower the oven temperature to 350 degrees and place the cookie sheet and dough, still covered by the cake pan into the oven. Bake for 15 minutes, then remove the cake pan. Continue baking about 15 to 20 minutes more, until deep golden and crisp, piercing the dough with a fork if it bubbles up. Cool the crust in the pan on a rack.
To assemble the tart, reheat the apricots for a few seconds over low heat to make sure they are not stuck to the pan. Invert the tart crust to a cardboard and slide the crust over the top of the pan the apricots are in. Cover the crust with the cardboard or a platter and invert the apricots onto the crust. Remove the pan and rearrange the apricots if necessary. Serve tepid with some whipped cream.
TIPS: If you are making puff pastry in the summer or a warm kitchen, chill all your ingredients as well as the food processor bowl and blade. The best dried apricots can be found in the health food section of your grocery store or in organic markets. Look for "unsulphered" apricots.